8. antennas and radiating systems · 8.1 introduction antennas radiation fundamentals antenna...

Dr. Rakhesh Singh Kshetrimayum

8. Antennas and Radiating Systems

Dr. Rakhesh Singh Kshetrimayum

4/26/20161 Electromagnetic Field Theory by R. S. Kshetrimayum

8.1 Introduction Antenna is a device used for radiating and receiving EM waves

Any wireless communication can’t happen without antennas.

Antennas have many applications like in mobile communications (all mobile phones has in-built antennas)

wireless local areas networks (your laptop connecting wireless

4/26/2016Electromagnetic Field Theory by R. S. Kshetrimayum2

wireless local areas networks (your laptop connecting wireless internet also has antennas)

television (old TV antennas are Yagi-Uda antennas, now, generally disc antennas are employed for direct to home (DTH) TVs)

satellite communications (usually have large parabolic antennas or microstrip antenna arrays)

rockets and missiles (microstrip antenna arrays)

8.1 Introduction

Antennas

Radiation fundamentals

Antenna pattern and parameters

Types of antennas

Antenna arrays

When does a charge radiate?


Fig. 8.1 Antennas (cover antenna pattern and parameters after types of antennas)

charge radiate?

Wave equation for potential functions

Solution of wave equation for potential functions

Hertz dipole

Dipole antenna

Loop antenna

8.2 Radiation fundamentals8.2.1 When does a charge radiates? accelerating/decelerating charges radiate EM waves

r


Fig. 8.2 A giant sphere of radius r with a source of EM wave at its origin

Source

r

8.2 Radiation fundamentals Consider a giant sphere of radius r which encloses the source

of EM waves at the origin (Fig. 8.2)

The total power passing out of the spherical surface is given by Poynting theorem,


( ) ( )∫∫ •×=•= ∗sdHEsdSrP avgtotal

rrrrrRe

( )lim

totalP P r

r=

→∝

8.2 Radiation fundamentals This is the energy per unit time that is radiated into infinity

and it never comes back to the source

The signature of radiation is irreversible flow of energy away from the source


from the source

Let us analyze the following three cases:

CASE 1: A stationary charge will not radiate. no flow of charge =>no current=>no magnetic field=>no

radiation

8.2 Radiation fundamentalsCASE 2: A charge moving with constant velocity will not radiate.

The area of the giant sphere of Fig. 8.2 is 4 r2

So for the radiation to occur Poynting vector must decrease no faster than 1/r2

from Coloumb’s law, electrostatic fields decrease as 1/r2,


from Coloumb’s law, electrostatic fields decrease as 1/r ,

whereas Biot Savart’s law states that magnetic fields decrease as 1/r2

So the total decrease in the Poynting vector is proportional to 1/r4 no radiation

CASE 3: A time varying current or acceleration (or deceleration) of charge will radiate.

8.2 Radiation fundamentals Basic radiation equation:

where L=length of current element, m

=time changing current, As-1(units)

dt

dvQ

dt

diL =

di


=time changing current, As-1(units)

Q=charge, C

=acceleration of charge, ms-2

dt

di

dt

dv

8.2 Radiation fundamentals

To create radiation there must be a time varying current or

acceleration (or deceleration) of charge

Static charges=>no radiation


Static charges=>no radiation

If the charge motion is time varying with acceleration or deceleration then there will be radiation even if the wire is straight

8.2 Radiation fundamentals

Charges moving with uniform velocity: no radiation if the wire is straight and infinite in extent and

radiation if the wire is curved,


curved,

bent,

discontinuous,

terminated or

truncated (these will either accelerate or decelerate the charge)

8.2 Radiation fundamentals For radiation, electric field will have a transversal component

instead of radial component

whenever a charge accelerates or decelerates

Example 8.1

Prove that in order to have radiation the electric field must


Prove that in order to have radiation the electric field must have spatial variation as 1/r where r is the distance from the source

8.2 Radiation fundamentals 8.2.2 Wave equation for potential functions

One of the Maxwell’s equation

Hence

Putting this in the following Maxwell’s equation

0=•∇ Br

ABrr

×∇=

B∂r

r ( )A×∇∂r

r


t

BE

∂

∂−=×∇

rr ( )

t

AE

∂

×∇∂−=×∇

rr

0A

Et

∂⇒∇× + =

∂

rr A

E Vt

∂⇒ + = −∇

∂

rr

8.2 Radiation fundamentals Putting this in the following Maxwell’s equation

ε

ρ=•∇ E

r

ε

ρ−=

∇+

∂

∂•∇ V

t

Ar

2AV

t

ρ

ε

∂⇒∇ • + ∇ = −

∂

r


Another Maxwell’s equation

Simplifies to

t ε∂

t

EJB

∂

∂+=×∇

rrr

µεµ

( )t

Vt

A

JAAA∂

∇+

∂

∂∂

−=∇−•∇∇=×∇×∇

r

rrrrµεµ2

8.2 Radiation fundamentals Lorentz Gauge condition

Applying above condition

0=∂

∂+•∇

t

VA µεr

∂ ∂ ∂r

r r


22

2

V AA J V

t t tµε µ µε

∂ ∂ ∂ ⇒∇ − − ∇ = − + ∇

∂ ∂ ∂

rr r

22

2

AA J

tµε µ

∂⇒∇ − = −

∂

rr r

( )ε

ρ−=∇+

∂

•∇∂V

t

A 2

r 22

2

VV

t

ρµε

ε

∂⇒∇ − = −

∂

8.2 Radiation fundamentals8.2.2 Wave equation for potential functions

From Maxwell’s equations for time varying fields, we have derived the two wave equations for potential functions (magnetic vector and electric potentials)

rr

r ∂ 2 2V ρ∂


Why find potential functions instead of fields?

Jt

AA

rr

rµµε −=

∂

∂−∇

2

22

22

2;

VV

t

ρµε

ε

∂∇ − = −

∂

8.2 Radiation fundamentals8.2.3 Solution of wave equation for potential

functions

For time harmonic functions of potentials,

JAArrr

µβ −=+∇ 22


where

To solve the above equation, we can apply Green’s function technique

Green’s function G is the solution of the above equation with the R.H.S equal to a delta function

( )spaceGG δβ =+∇ 22

µεωβ =

8.2 Radiation fundamentals Once we obtain the Green’s function,

we can obtain the solution for any arbitrary current source by applying the convolution theorem

For radiation problems, the most appropriate coordinate system is spherical


the most appropriate coordinate system is spherical since the wave travels out radially in all directions

It has also symmetry along θ and directions

Hence, the above equation reduces to

0G G

θ φ

∂ ∂= =

∂ ∂

( )rGr

Gr

rrδβ =+

∂

∂

∂

∂ 22

2

1

8.2 Radiation fundamentals Putting Ψ= G r,

For r not equal to 0,

( )rrr

δβ =Ψ+Ψ∂

∂ 2

2

2


Therefore,

02

2

2

=Ψ+Ψ∂

∂β

r

rjrjBeAe

ββ +− +=Ψ

8.2 Radiation fundamentals Since the radiation travels radially in positive r direction

negative r direction is not physically feasible for a source of a field, we get,

rjAe

β−=Ψ

rjβ−


From example 8.2, we can find the constant A and hence

r

AeG

rjβ−

=

r

eGA

rj

ππ

β

4;

4

1 −

==

8.2 Radiation fundamentals Since the medium surrounding the source is linear,

we can obtain the potential for any arbitrary current input by the convolution of the impulse function (Green’s function) with the

input current

( )'rrj −−

r

rrβ

µ


( ) '

'

'0

4dv

rr

erJA

V

rrj

∫−

=

−−

rr

rrβ

π

µ

8.2 Radiation fundamentals The prime coordinates denote the source variables

unprimed coordinates denote the observation points

The modulus sign in is to make sure that

is positive

since the distance in spherical coordinates is always positive

'rr −


since the distance in spherical coordinates is always positive

8.3 Antenna pattern and parameters

8.3.1 What is antenna radiation pattern?

The radiation pattern of an antenna is a 3-D graphical representation of the radiation properties of the antenna as a function of position (usually in spherical coordinates)

If we imagine an antenna is placed at the origin of a spherical


If we imagine an antenna is placed at the origin of a spherical coordinate system, its radiation pattern is given by measurement of the magnitude

of the electric field over a surface of a sphere of radius r


Dipole AntennaOmni-directional radiation pattern


Horn Antenna Directional radiation pattern


For a fixed r, electric field is only a function of θ and

Two types of patterns are generally used: (a) field pattern (normalized or versus spherical

coordinate position) and

φ

( ),E θ φr

Er

Hr


coordinate position) and

(b) power pattern (normalized power versus spherical coordinate position).


3-D radiation patterns are difficult to draw and visualize in a 2-D plane like pages of this book

Usually they are drawn in two principal 2-D planes which are orthogonal to each other

Generally, xz- and xy- plane are the two orthogonal principal


Generally, xz- and xy- plane are the two orthogonal principal planes

E-plane (H-plane) is the plane in which there are maximum electric (magnetic) fields for a linearly polarized antenna



Fig. 8.3 (c) Typical radiation pattern of an antenna

maxθ

θ


A typical antenna radiation pattern looks like as in Fig. 8.3 (c)

It could be a polar plot as well

An antenna usually has either one of the following patterns: (a) isotropic (uniform radiation in all directions, it is not

possible to realize this practically)


possible to realize this practically)

(b) directional (more efficient radiation in one direction than another)

(c) omnidirectional (uniform radiation in one plane)



Fig. 8.3 (a) Antenna field regions

3

max10 0.62nf

Dr

λ< ≤

3 2

max max2

20.62 nf

D Dr

λ λ< ≤

2

max2ff

Dr

λ<


The antenna field regions could be divided broadly into three regions (see Fig. 8.3 (a)):

Reactive near field region:

This is the region immediately surrounding the antenna where the reactive field (stored energy-standing waves)


where the reactive field (stored energy-standing waves) dominates

Reactive near field region is for a radius of

where Dmax is the maximum antenna dimension

3

max10 0.62

nf

Dr

λ< ≤


Radiating near field (Fresnel) region:

The region in between the reactive near field and the far-field (the radiation fields are dominant)

the field distribution is dependent on the distance from the antenna


antenna

Radiating near field (Fresnel) region is usually for a radius of

3 2

max max2

20.62 nf

D Dr

λ λ< ≤


Far field (Fraunhofer) region:

This is the region farthest from the antenna where the field distribution is essentially independent of the distance from the antenna (propagating waves)

Fraunhofer far field region is usually for a radius of2

max2ff

Dr

<


Fraunhofer far field region is usually for a radius of

In the far field region, the spherical wavefront radiated from a source antenna can be approximated as plane wavefront

The phase error in approximating this is π/8

maxffr

λ<


Fig. 8.3 (b) Illustration of

far field region

(antenna under test: AUT)



We can calculate the distance rff by equating the maximum error (which is at the edges of the AUT of maximum dimension Dmax) in the distance r by approximating spherical wavefront to plane wavefront to λ/16 (Exercise 8.1)

8.3.2 Direction of the main beam (θmax)


8.3.2 Direction of the main beam (θmax)

A radiation lobe is a clear peak in the radiation intensity surrounded by regions of weaker radiation intensity


Main beam is the biggest lobe in the radiation pattern of the antenna

It is the radiation lobe in the direction of maximum radiation

θmax is the direction in which maximum radiation occurs

Any lobe other than the main lobe is called as minor lobe


Any lobe other than the main lobe is called as minor lobe

The radiation lobe opposite to the main lobe is also termed as back lobe This will be more appropriate for polar plot of radiation pattern


8.3.3 Half power beam width (HPBW)

It is the angular separation between the half of the maximum power radiation in the main beam

At these points, the radiation electric field reduces by of the maximum electric field

1

2


of the maximum electric field

Half power is also equal to -3-dB

We also call HPBW as -3-dB beamwidth

They are measured in the E-plane and H-plane radiation patterns of the antenna


8.3.4 Beam width between first nulls (BWFN)

It is the angular separation between the first two nulls on either side of the main beam

For same values of BWFN, we can have different values of HPBW for narrow beams and broad beams


HPBW for narrow beams and broad beams

HPBW is a better parameter for specifying the effective beam width


8.3.5 Side lobe level (SLL)

The side lobes are the lobes other than the main beam and it shows the direction of the unwanted radiation in the antenna

radiation pattern

The amplitude of the maximum side lobe in comparison to


The amplitude of the maximum side lobe in comparison to the main beam maximum amplitude of the electric field is called as side lobe level (SLL)

It is normally expressed in dB and a SLL of -30 dB or less is considered to be good for a communication system


8.3.6 Radiation intensity

Power crossing the area dA is

Radiation intensity is defined as power crossing per unit solid angle

( )dAS φθ ,

( ) ( ) ( ) ( )SrWrSdAS

U /,,

, 2φθφθ

φθ ==


2

4 0 0

( , ) ( , ) sinradP U d U d d

π π

π θ φ

θ φ θ φ θ θ φΩ= = =

= Ω =∫ ∫ ∫

2

0 0

1( , )sin

4 4

radavg

PU U d d

π π

θ φ

θ φ θ θ φπ π

= =

= = ∫ ∫

( ) ( ) ( )SrWrSd

U /,, φθφθ =Ω

=


8.3.7 Directivity

The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction from the

antenna

to the radiation intensity averaged over all directions


to the radiation intensity averaged over all directions which equivalent to the radiation intensity of an isotropic antenna

2

0 0

( , ) 4 ( , )( , )

( , )sinavg

U UD

UU d d

π π

θ φ

θ φ π θ φθ φ

θ φ θ θ φ= =

= =

∫ ∫


D (θ, ) is maximum at θmax and minimum along θnull

is also known as beam solid angle

φ

( ) max maxmax 2

0 0

4 ( , ) 4,

( , )sin A

U UD

U d d

π π

θ φ

π θ φ πθ φ

θ φ θ θ φ= =

= = =Ω

∫ ∫avgU

Ω


is also known as beam solid angle

It is also defined as the solid angle through which all the antenna power would flow if the radiation intensity was for all angles in

AΩ

max ( , )U θ φ AΩ


Given an antenna with one narrow major beam, negligible radiation in its minor lobes

where and are the half-power beam widths in

HPBW HPBW

rad rad

A θ φΩ ≈ ×

θ φ


where and are the half-power beam widths in radians which are perpendicular to each other

For narrow beam width antennas

It can be shown that the maximum directivity is given by

HPBWθHPBWφ

( ), 1HPBW HPBW

θ φ <<

max

4

HPBW HPBW

rad radD

π

θ φ≅

×


If the beam widths are in degrees, we have

8.3.8 Gain

2

max deg deg deg deg

1804

41,253

HPBW HPBW HPBW HPBW

D

ππ

θ φ θ φ

≅ =

× ×


8.3.8 Gain

In defining directivity, we have assumed that the antenna is lossless

But, antennas are made of conductors and dielectrics


It has same in-built losses accompanied with the conductors and dielectrics

Thereby, the power input to the antenna is partly radiated and remaining part is lost in the imperfect conductors as well as in


remaining part is lost in the imperfect conductors as well as in dielectrics

The gain of an antenna in a given direction is defined as the ratio of the intensity in a given direction

to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically


Note that definitions of the antenna directivity and gain are essentially the same

( ) radrad

4 ( , )e4 ( , ), e ( , )

input rad

UUG D

P P

π θ φπ θ φθ φ θ φ= = =


essentially the same except for the power terms used in the definitions

Directivity is the ratio of the antenna radiated power density at a distant point to the total antenna radiated power radiated isotropically

Gain is the ratio of the antenna radiated power density at a distant point to the total antenna input power radiated isotropically


The antenna gain is usually measured based on Friistransmission formula and it requires two identical antennas

One of the identical antennas is the radiating antenna, and the other one is the receiving antenna

Assuming that the antennas are well matched in terms of


Assuming that the antennas are well matched in terms of impedance and polarization, the Friis transmission equation is

2

10 10

1 420log 10log

4 2

r rt r t r

t t

P PRG G G G G G

P R P

λ π

π λ

= = = ∴ = +

Q


Friis transmission equation states that the ratio of the received power at the receiving antenna and transmitted power at the transmitting antenna is: directly proportional to both gains of the transmitting (Gt) and

receiving (Gr) antennas


receiving (Gr) antennas

inversely proportional to square of the distance between the transmitting and receiving antennas (1/R2) and

directly proportional to the square of the wavelength of the signal transmitted (λ2)


Assumptions made are:

(a) antennas are placed in the far-field regions

(b) there is free space direct line of sight propagation between the two antennas

(c) there are no interferences from other sources and


(c) there are no interferences from other sources and no multipaths between the transmitting and receiving antennas

due to reflection,

refraction and

diffraction


8.3.9 Polarization

Let us consider antenna is placed at the origin of a spherical coordinate system and wave is propagating radially outward in all directions

In the far field region of an antenna,


In the far field region of an antenna,

( ) ( ) ( )ˆ ˆ, , ,E E Eθ φθ φ θ φ θ θ φ φ= +r


Putting the time dependence, we have,

where δ is the phase difference between the elevation and azimuthal components of the electric field

( ) ( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆ, , , cos , cos , , , ,E t E t E t E t E tθ φ θ φθ φ θ φ ω θ θ φ ω δ φ θ φ θ θ φ φ= + + = +r


azimuthal components of the electric field

The figure traced out by the tip of the radiated electric field vector as a function of time for a fixed position of space can be defined

as antenna polarization


a) LP

When δ=0, the two transversal electric field components are in time phase

The total electric field vector


makes an angle with the -axis

( ) ( ) ( ) ( ) ( )( )ˆ ˆ, , , cos , cosE t E t E tθ φθ φ θ φ ω θ θ φ ω φ= +r

LPθ θ

( )( )

( )( )

1 1, , , ,

tan tan, , , ,

LP

E t E t

E t E t

φ φ

θ θ

θ φ θ φθ

θ φ θ φ− −

= =


The tip of the total radiated electric field vector traces out a line

Therefore, the antenna’s polarization is LP

b) CP

When , the two transversal electric field components πδ = ±


When , the two transversal electric field components are out of phase in time

and if the two transversal electric field components are of equal amplitude

2

πδ = ±

( ) ( ) ( )0, , ,E E Eθ φθ φ θ φ θ φ= =


The total electric field vector

makes an angle with the -axis

( ) ( ) ( ) ( ) ( )( )ˆ ˆ, , , cos , sinE t E t E tθ φθ φ θ φ ω θ θ φ ω φ= +r

CPθ θ


This implies that the total radiated electric field vector of the antenna traces out a circle as time progresses from 0 to 2 and so on

( )( )

( )1 1sin

tan tan tancos

CP

tt t

t

ωφ ω ω

ω− −

= = =

π

ω


If the vector rotates counterclockwise (clockwise), then the antenna polarization is RHCP (LHCP)

For

the total electric field vector traces out an ellipse and

( ) ( ), , 0,E E andθ φθ φ θ φ α π≠ ≠ πδ ,0≠


the total electric field vector traces out an ellipse and hence it is elliptically polarized (EP)

The ratio of the major and minor axes of the ellipse is called axial ratio (AR)

For instance, AR=0 dB for CP and AR= ∞ dB for LP


Let us try to understand two terms (co- and cross-polarization) which are important for the antenna radiation pattern

Co-polarization means you measure the antenna with another antenna oriented in the same polarization with the antenna


antenna oriented in the same polarization with the antenna under test (AUT)

Cross-polarization means that you measure the antenna with antenna oriented perpendicular w.r.t. the main polarization


Cross-polarization is the polarization orthogonal to the polarization under consideration

For example, if the field of an antenna is horizontally polarized, the cross-polarization for this case is vertical polarization


polarization

If the polarization is RHCP, the cross-polarization is LHCP

Let us put this into mathematical expressions:

We may write the total electric field propagating along z-axis as

( )ˆ ˆ j z

co co cr crE E u E u e

β−= +r


where the co- and cross-polarization unit vectors satisfy the orthonormality condition

Therefore, the co- and cross-polarization components of the

* * * *ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1, 1, 0, 0co co cr cr co cr cr co

u u u u u u u u• = • = • = • =


Therefore, the co- and cross-polarization components of the electric fields can be obtained as

* *ˆ ˆ,co co cr crE E u E E u= • = •r r


a) LP

For a general LP wave, we can write,

For a x-directed LP wave, =0, hence,

ˆ ˆ ˆ ˆ ˆ ˆcos sin , sin cosco LP LP cr LP LP

u x y u x yφ φ φ φ= + = −

LPφ


For a y-directed LP wave, =900, hence,

* *ˆ ˆ ˆ ˆ ˆ ˆ, ; ,co cr co co x cr cr y

u x u y E E u E E E u E= = − = • = = • = −r r

LPφ

* *ˆ ˆ ˆ ˆ ˆ ˆ, ; ,co cr co co y cr cr xu y u x E E u E E E u E= = = • = = • =r r


b) CP

For a RHCP wave, we can write,

ˆ ˆ ˆ ˆˆ ˆ,

2 2co cr

x jy x jyu u

− += =


For a LHCP wave, co- and cross-polarization unit vectors and components of the electric field will interchange

* *ˆ ˆ,2 2

x y x y

co co cr cr

E jE E jEE E u E E u

+ −= • = = • =r r


c) EP

For a EP wave, we can write,

2 2

ˆ ˆ ˆ ˆˆ ˆ,

1 1

EP EPj j

co cr

x Ae y Ae x yu u

A A

φ φ−+ − += =

+ +


In order to determine the far-field radiation pattern of an AUT, two antennas are required

The one being tested (AUT) is normally free to rotate and it is connected in receiving mode


Note that AUT as a receiving antenna measurement will generate the same radiation pattern to that of AUT used as a transmitting antenna (from reciprocity theorem)

Another antenna is usually fixed and it is connected in transmitting mode


transmitting mode

The AUT is rotated by a positioner and it can rotate 1-, 2-and 3-degrees of freedom of rotation


The AUT is rotated in usually two principal planes (elevation and azimuthal)

The received field strength is measured by a spectrum analyzer or power meter which will be used to generate the antenna radiation pattern in


which will be used to generate the antenna radiation pattern in two principal planes also known as E- and H- planes

The antenna radiation patterns in these two principal planes can be used to generate the 3-D radiation pattern of an antenna



Equivalent circuit of an antenna


8.3.10 Quality factor and bandwidth

The equivalent circuit of a resonant antenna can be approximated by a series RLC resonant circuit

where R=Rr+RL are the radiation and loss resistances,


L is the inductance and

C is the capacitance of the antenna


For a resonant antenna like dipoles, the FBW is related to the radiation efficiency and quality factor Q (FBW=1/Q)

The quality factor of an antenna is defined as 2πf0 (f0 is the resonant frequency) times the energy stored over the


the resonant frequency) times the energy stored over the power radiated and Ohmic losses

( )

( ) ( )

( )

2 2

2

0 00

20

0

1 1 1

4 4 2 2 12

1 2

2

1

2

r L r Lr L

rlossless rad

r r L

I L If L f L

Q fR R f R R C

I R R

RQ e

f R C R R

π ππ

π

π

+

= = =+ ++

= = •+


where Qlossless is the quality factor when the antenna is lossless (RL=0) and

erad is the antenna radiation efficiency.

Note that the radiation efficiency of an antenna is defined as the ratio of the power delivered to the radiation resistance Rr


the ratio of the power delivered to the radiation resistance Rr

to the power delivered to Rr and RL

( ) ( )

2

2

1

21

2

rr

rad

r Lr L

I RR

eR R

I R R

= =++



Small Antennas


There is a very important concept on designing electrically small antennas

When kr<1 (electrically small antennas), the quality factor Q of a small antenna can found from the J. L. Chu’s relation

( )2

+


where k is the wave number and r is the radius of the smallest sphere enclosing the antenna

( )

( ) ( )

2

3 2

1 2

1rad

krQ e

kr kr

+= •

+


The above relation gives the relationship between the antenna size, efficiency and quality factor.

This expression can be reduced further for smallest Q for a LP very small antenna (kr<<1) as follows:


Harrington gave also a practical upper limit to the gain of a small antenna for a reasonable BW as

( )min 3

1 1Q

krkr= +

( ) ( )2

max 2G kr kr= +


For example, for a Hertz dipole of very small length 0.01λ, Qmin =32283

It has very high Q and hence a very narrow FBW (0.000031)

Gmax=0.0638 or -12dB

Note that in the above calculations r=0.005λ has been used


Note that in the above calculations r=0.005λ has been used

But the gain of the antenna in dB is also negative

Antenna size, quality factor, bandwidth and radiation efficiency is interlinked

There is no complete freedom to optimize each one of them independently

8.3 Antenna pattern and parametersRECAP:

Antenna acts as an interface between a guided wave and a

free-space wave

One of the most important characteristics of an antenna is its

directional property

Ability to concentrate radiated power in a certain directionAbility to concentrate radiated power in a certain direction

Or receive power from a preferred direction

This directional property is characterized in Radiation pattern

From reciprocity theorem, we can show that

the pattern characteristics of an antenna are the same in the

transmit and

the receive modes


FeedLine Antenna

Source

Radiated fields


A plot of electric or magnetic field intensity as function of the direction at a constant distance from the antenna is known as the electric field pattern or magnetic field pattern

The field intensity along a direction (θ,φ) is given by the length of the position vector to a point on the surface of the 3D shape in the direction (θ,φ)


8.3 Antenna pattern and parameters Directivity of an isotropic antenna

is taken for reference for defining gain

Directivity of an antenna indicates how well the antenna radiate in a particular direction in comparison with an isotropic antenna

P0/4π


with an isotropic antenna radiating the same amount of power (P0)

Assume erad=1,

gain = directivity

DdB=10log10(D) dBi

2

0 0

( , ) 4 ( , )( , )

( , )sinavg

U UD

UU d d

π π

θ φ

θ φ π θ φθ φ

θ φ θ θ φ= =

= =

∫ ∫


Generally single value of gain or directivity is given in that case the gain or directivity is along the main beam peak

Antenna equivalent circuit can be thought of as Radiation resistance

Loss resistance


Loss resistance

Reactive parts

So the feed line also has a characteristic impedance Z0 usually of 50 Ohm

At the input of the antenna, the impedance seen by the feed line can be assumed as ZL


Then the reflection coefficient and VSWR may be calculated as

2:

1,10

1

1;

0

0

≤

∞≤≤≤Γ≤

Γ−

Γ+=

+

−=Γ

VSWRBW

VSWR

VSWRZZ

ZZ

L

L


Polarization of the antenna is the polarization of the wave radiated by the antenna in the far field

Polarization of the antenna is direction-dependent

Conventionally polarization of the wave is along the main beam direction


Received signal power in a wireless communication link

The received signal power is a function of four elements: The transmitter output electrical power (pT)

The fraction of the transmitter electrical power directed at the receiver defined by the transmit antenna gain (gT)


receiver defined by the transmit antenna gain (gT)

The loss of energy in the communication medium

The fraction of the received electrical power directed at the receiver defined by the received antenna gain (gR)


Power flux density (pfd) for isotropic radiator at a distance r =p0/(4πr2)

Power transmitted= pT gT

pfd= pT gT /(4πr2)

In dB,


In dB,

PFD= PT +GT-10log4πr2=EIRP-10log4πr2 (dBW/m2)

where equivalent isotropic radiated power=EIRP

No matter what, a large portion of the transmitted energy 10log4πr2 is not seen by the receiver


The received power at the receiver antenna is

Pr=pfd×Aer

Effective aperture is defined as the ratio of power delivered to the load if antenna to the Poynting vector in any direction (θ,φ)


(θ,φ)

Ae (θ,φ) =Pr (θ,φ) /S (m2)

Note that for any antenna it can be shown that

G/Ae=(4×π)/λ2

Or, Ae=G λ2 /(4×π)


Pr=pfd×Ae= pT gTgr λ2 /(4×πr)2

PR=PT+GT+GR-20log((4×πr) ×c/f) (dBW)

The last term is called range loss or power loss between two isotropic antennas at particular range and frequency


8.4 Kinds of antennas8.4.1 Hertz dipole

Let us find the fields of a small current carrying element of length dl

An infinitesimally small current element is called a Hertz dipole


Hertz dipole

Electrically small antennas are small relative to wavelength

Whereas electrically large antennas are large relative to wavelength

Hertz dipole is not of much practical use but it is the basic building block of any kind of antennas

8.4 Kinds of antennas The infinitesimal time-varying current in the Hertz dipole is

where ω is the angular frequency of the current

Since the current is assumed along the z-direction, the

( ) zeItI tj ˆ0ω=


Since the current is assumed along the z-direction, the magnetic vector potential at the observation point P is along z-direction (see section 3.10)

zr

edleIzAA

tjrj

z ˆ4

ˆ 00

π

µ ωβ−

==r

8.4 Kinds of antennas Fig. 8.5 Hertz dipole

located at the origin and

oriented along z-axis ( ), ,P r θ φ

r

θ

φ


0

j tI e

ω

8.4 Kinds of antennas Note that for this case

For infinitesimally small current element at the origin

using the coordinate transformation from the Rectangular to Spherical coordinate systems (see section 1.3.3)

( )' '

0

j tJ r dv I dle

ω=r r

'r r r− =r r r


Spherical coordinate systems (see section 1.3.3)

sin cos sin sin cos sin cos sin sin cos 0

cos cos cos sin sin cos cos cos sin sin 0

sin cos 0 sin cos 0

xr

y

zz

AA

A A

AA A

θ

φ

θ φ θ φ θ θ φ θ φ θ

θ φ θ φ θ θ φ θ φ θ

φ φ φ φ

= − = − − −

cos ; sin ; 0r z zA A A A Aθ φθ θ⇒ = = − =

8.4 Kinds of antennas

Using the symmetry of the problem (no variation in ), we

0

AH

µ

∇×=

rr

Q2

0

ˆ ˆˆ sin

1

sin

sinr

r r r

r r

A rA r Aθ φ

θ θφ

µ θ θ φ

θ

∂ ∂ ∂ = ∂ ∂ ∂

φ


Using the symmetry of the problem (no variation in ), we have,

2

0

ˆ ˆˆ sin

10

sin

cos sin 0z z

r r r

Hr r

A rA

θ θφ

µ θ θ φ

θ θ

∂ ∂ ∂ = ≡ ∂ ∂ ∂

−

r

φ


( ) ( )2

0

sin0; 0; sin cos

sinr z z

rH H H rA A

r rθ φ

θθ θ

µ θ θ

∂ ∂ ⇒ = = = − −

∂ ∂

( )0 0ˆ ˆcos sin

sin sin4 4

j t j tj r j rj r j rI dle I dlee e

H e j er r r r r

ω ωβ ββ β

φ

φ φθ θθ β θ

π θ π

+ +− −− −

∂ ∂ ⇒ = − − = +

∂ ∂


The Hertz dipole has only component of the magnetic field, i.e., the magnetic field circulates the dipole

0

2

ˆ sin 1

4

j t j rI dle e j

r r

ω βφ θ β

π

+ − = +

φ

8.4 Kinds of antennas The electric field for ( in free space, we don’t have

any conduction current flowing) can be obtained as

Using the symmetry of the problem (no variation in ) like

0J =r

ωεj

HE

rr ×∇

=

φ


Using the symmetry of the problem (no variation in ) like before, we have,

φ

( ) ( )2 2

ˆ ˆˆ sin

1 1 ˆˆ0 sin sinsin sin

0 0 sin

r r r

E r H r r H rj r r j r r

r H

φ φ

φ

θ θφ

θ θ θωε θ θ φ ωε θ θ

θ

∂ ∂ ∂ ∂ ∂ = ≡ = − ∂ ∂ ∂ ∂ ∂

r


Note that Er has only 1/r2 and 1/r3 variation with r

( )20 0

2 2 2 2

1 1sin 2sin cos

sin 4 sin 4

j t j r j t j r

r

I dle e I dle ej jE r r

j r r r j r r r

ω β ω ββ βθ θ θ

ωε θ π θ ωε θ π

− −∂ = + = +

∂

( )

+∂

∂−=

∂

∂−= − rj

tj

er

jrrj

dleIHr

rrj

rE

βω

φθ βπωε

θθ

θωε

1

4

sinsin

sin

0

2


We see that electric field is in the (r, θ) plane whereas the magnetic field has component only

rj θωε sin

2

0

2 3

sin

4

j tj rI dle j j

er r r

ωβθ β β

πε ω ω ω−

= + −

φ

8.4 Kinds of antennas Therefore, the electric field and magnetic field are

perpendicular to each other (Which wave is this?)

Points to be noted:

1) Fields can be classified into three categories

(a) Radiation fields (spatial variation 1/r)


(a) Radiation fields (spatial variation 1/r)

(b) Induction fields (spatial variation 1/r2) and

(c) Electrostatic fields (spatial variation 1/r3)

8.4 Kinds of antennas 2) Field variation with f since

a) Electrostatic fields (1/r3) are also inversely proportional to the frequency ( )

b) Induction field (1/r2) is independent of frequency ( )

µεωβ =

β

ω

1


b) Induction field (1/r2) is independent of frequency ( )

c) Radiation field (1/r) is proportional to frequency ( )ω

β 2

ω

β

8.4 Kinds of antennas 2) Field variation with r

For small values of r, electrostatic field is the dominant term and

For large values of r,

radiation field is the dominant term


radiation field is the dominant term

We can also observe that the three types of fields are equal in magnitude when

β2/r= β/r2=1/r3 => r=1/β= λ/2

8.4 Kinds of antennas For r< λ/2, 1/r3 term dominates

For r>> λ/2, the 1/r term dominates

Near field region:

For r<< λ/2

(in fact the near field region distance r= λ/2 is for D<<λ


(in fact the near field region distance r= λ/2 is for D<<λfor an ideal infinitesimally small Hertz dipole

and the near field region distance for is for D>>λ), electrostatic fields dominate

3

0.62D

rλ

=

8.4 Kinds of antennas as r<< λ/2

1→− rje

βQ

0

3

cos2 ;

4

j t

r

I dl eE j

r

ωθ

πεω≈ − 0

3

sin

4

j tI dl eE j

r

ω

θ

θ

πεω≈ −


The magnitude of the near field is

4 rπεω 4 rπεω

2 2 2 20

34cos sin

4r

I dlE E E

rθ θ θ

πεω= + = +

8.4 Kinds of antennas A polar plot of the near field can be generated by writing a

MATLAB program for plotting

Maximum field is along θ=00, θ=1800 and minimum is along

( ) θθθ 22 sincos4 +=F


Maximum field is along θ=00, θ=1800 and minimum is along θ=900, θ=2700 (see Fig. 8.6(a))



Fig. 8.6 (a) Near field pattern plot of a Hertz dipole located at the origin and oriented along z-axis (maximum radiation along z-axis)

8.4 Kinds of antennas Far field region:

For r>> λ/2 (in between reactive near field and Fraunhofer far field region, there exists the Fresnel near field region that’s why we have chosen an r>> λ/2), radiation field is the dominant term


field is the dominant term

In other words kr>>1, we have,

2

0 0sin sin;

4 4

j t j r j t j rI dl e e I dl e eE j H j

r r

ω β ω β

θ φ

θ β θ β

πεω π

− −

= =

8.4 Kinds of antennas The electric fields and magnetic fields are in phase with each

other

They are 90˚ out of phase with the current due to the (j) term in the expressions of Eθ and Hφ

It is interesting to note that the ratio of electric field and


It is interesting to note that the ratio of electric field and magnetic field is constant

E

H

θ

φ

ω µεβ µη

ωε ωε ε= = = =

8.4 Kinds of antennas Hence, the fields have sinusoidal variations with θ

They are zero along θ=0

No radiation along z-axis unlike near field case

They are maximum along θ=π /2 (see Fig. 8.6 (b))




Fig. 8.6 (b) E-plane radiation pattern of a Hertz dipole in far field (H-plane radiation will look like a circle)

8.4 Kinds of antennas Power flow:

( ) * *1 1 ˆ ˆˆRe Re2 2

avg rS E H E r E Hθ ϕθ φ= × = + ×r r

*1Re

2avgS E H rθ φ= $

2 3

0 sin1 I dlr

θ β =

$


Antenna power flows radially outward Power density is not same in all directions The net real power is only due to the radiations fields (i.e.

jβ2/r and jβ/r) of electric and magnetic fields

Re2

avgS E H rθ φ=2 4

rrπ ωε

=

$

8.4 Kinds of antennas In the far field, electric field is along direction

magnetic field is along direction

Both of them are perpendicular to the power flow (Poyntingvector) which is along direction

Total radiated power:

θ

φ

r$


Total radiated power:

The total radiated power from a Hertz dipole

2 sinavgW S r d dθ θ φ= ∫∫2

2 2

040dl

W Iπλ

∴ =

8.4 Kinds of antennas Power radiated by the Hertz dipole is proportional to

the square of the dipole length and

inversely proportional to the dipole wavelength

It implies more and more power is radiated as the frequency and


the frequency and

the length

of the Hertz dipole increases

Radiation resistance of a Hertz Dipole:

Hertz dipole can be equivalently modeled as a radiation resistance

8.4 Kinds of antennasSince W=1/2 I0

2 Rrad

implies that Rrad = 80π2

Radiation pattern of a Hertz Dipole:

2

2 2

040dl

W Iπλ

∴ =

2dl

λ


Radiation pattern of a Hertz Dipole:

F(θ, )=sin θ for a Hertz dipole

The 3D plot of sin θ looks like an apple (see Figure 8.6 (c))



Fig. 8.6 (c) A typical 3-D radiation pattern of a Hertz dipole in the far field

8.4 Kinds of antennas To get the 3-D plot from the 2-D plot you need to rotate the

E-plane pattern along the H-plane pattern

For this case it will give the shape of an apple

Note that θ is also known as elevation angle and as azimuth angle

φ


azimuth angle

E-plane pattern for a dipole is also known as elevation pattern

H-plane pattern as azimuthal pattern

8.4 Kinds of antennas As mentioned before, it is easier to visualize 2-D plots than

3-D on a 2-D plane like pages of this book

Two principal planes radiation patterns are normally plotted E-plane (vertical: all planes containing z-axis like xz-plane, yz-

plane)


plane)

H-plane (horizontal) radiations patterns

are sufficient to describe the radiation pattern of a Hertz dipole

H-plane (xy-plane) radiation pattern is in the form of circle of radius 1 since F(θ, ) is independent of φ φ

8.4 Kinds of antennas Since the ratio of electric and magnetic field amplitudes in

the far field region of an antenna is 377 Ohm We may also plot the power pattern instead of field pattern

from the relation p(θ, )=sin2 θ

Polarization of Hertz dipole:


In the far-field region of a Hertz dipole, only θ component of the electric field the electric field is LP along direction

That means electric field is perpendicular to the line from the center of the dipole to the field observation point (see Fig. 8.5) and it lies in the plane containing this line and the dipole axis

θ

8.4 Kinds of antennas8.4.2 Dipole antenna

The next extension of a Hertz dipole is a linear antenna or a dipole of finite length as depicted in Fig. 8.7 (a)

It consists of a conductor of length 2L fed by a voltage or current source at its center


current source at its center

The current distribution can be obtained by assuming an o.c. transmission line

For o.c. transmission line

( )0 0

0 0

( ) 2 sinj z j zV VI z e e j z

Z Z

β β β+ +

− += − = −


θ


Fig. 8.7 (a) Dipole of length

2L (Dipole can be assumed

to composed of many Hertz dipoles)

8.4 Kinds of antennas The current is zero at z= L (since at the ends, there is no

path for the current to flow), so, we can write,

The electric field due to the current element dz

( )0 00 0

0 0

( ) 2 sin( ( )) sin ; 2V V

I z j L z I L z I jZ Z

β β+ +

= − − = − = −


The electric field due to the current element dz (it has the same expression of the Hertz dipole of the previous section except that now we have a length of dz and current of I(z) )

at far away observation point or in the far field can be written as

12

1 0

sin ( );

4

j R dEj I z dzedE dH

R

βθ

θ φ

β θ

πεω η

−

= =

8.4 Kinds of antennas Since the observation point P is at a very far distance, the

lines OP and QP are parallel and therefore

Note that for the amplitude, we can approximate 1 cosR R z θ∴ ≅ −

1 1≅


since the dipole size is quite small in comparison to the distance of the observation point P from the origin

Hence

1

1 1

R R≅

2 cossin ( )

4

j R j Zj I z dze e

dER

β β θ

θ

β θ

πεω

−

≅

8.4 Kinds of antennas Since we have assumed that the dipole of length 2L is

composed of many Hertz dipoles as depicted in Fig. 8.7 (a) (this is one of the reasons why we say that Hertz dipole or

infinitesimal dipole is the building block for many antennas),

we can write the total radiated electric field as


we can write the total radiated electric field as

2 cos2 cos0sin sin( ( ))sin ( )

4 4

j R j ZL L Lj R j Z

z L z L z L

j I L z e ej I z e eE dE dz dz

R R

β β θβ β θ

θ θ

β θ ββ θ

πεω πεω

−−

=− =− =−

−= = =∫ ∫ ∫

8.4 Kinds of antennas It can be shown that (see textbook for derivations)

In the previous equation, the term under bracket is F(θ)

and it is the variation of electric field as a function of θ and

( )( )0 0

cos cos cos60 60

sin

j R j RL Le eE j I j I F

R R

β β

θ

β θ βθ

θ

− − −⇒ ≅ =


and it is the variation of electric field as a function of θ and

it is the E-plane radiation pattern

8.4 Kinds of antennas In the H-plane,

Eθ is a constant and it is not a function of

hence it is a circle

The E–plane radiation pattern of the dipole varies with the length of the dipole as depicted in Fig. 8.8

φ


Note that according to Fig. 8.7 (a), we have considered the total dipole length is 2L) and the H-plane radiation is always a circle

Fig. 8.8 E-plane radiation pattern for dipole of length

(a) 2L=2×λ/4= λ/2

(b) 2L=2×λ=2λ

(c) 2L=2×2λ=4λ



(a) 2L=2×λ/4= λ/2



(b) 2L=2×λ=2λ



(c) 2L=2×2λ=4λ

8.4 Kinds of antennasPoints to be noted:

1. Input impedance of the dipole (z=0)

For dipole of length 2L, where L =odd multiples of

0 sin

in inin

in

V VZ

I I Lβ= =


For dipole of length 2L, where L =odd multiples of

=1,

For dipole of length 2L, where L =even multiples of .

, , sin4 2

mL L

λ πβ β=

0

inin

VZ

I=

, , sin 04

L m Lλ

β π β= = inZ⇒ = ∞

8.4 Kinds of antennas That’s why it is preferable to have dipoles of length odd multiples

of λ/2,

otherwise it is difficult to have a source with infinite impedance

2. Since increasing the dipole length more and more current is available for radiation,


available for radiation, • the total power radiated increases monotonically

3. The electric field has only component and hence it is linearly polarized

4. The radiation pattern have nulls and it can be calculated by equating F(θ)=0

$θ


cos( cos ) cos0

sin

null

null

L Lβ θ β

θ

−=

nullθcos⇒ 1mλ

π= ± ±


o For m=0,

But, in denominator is also zero

So let us take the limit of F(θ) as θ→0, and see

πθ θ ,0cos ,1 =±= nullnull

sinnull

θ

8.4 Kinds of antennas( ) ( ) ( ) ( ) ( ) ( )

0, 0,

2 4 2 4cos cos cos cos cos1

1 1sin sin 2! 4! 2! 4!

L L L L L L

Lim Limθ π θ π

β θ β β θ β θ β β

θ θ→ →

− ≅ − + − − +

( ) ( ) ( ) ( ) ( ) ( )0, 0,

4 42 24 22 1 cos sin 1 cossin sin10

2! 4! sin 2! 4!

L LL L

Lim Limθ π θ π

β θ β θ θβ θ β θ

θ→ →

− + = − × = − =


5. To find θ for maximum radiation, we have to find the solution of

=0

We can also take the mean of the first two nulls to approximate

( )dF

d

θ

θ

maxθ

8.4 Kinds of antennasMonopole antennas:

A monopole is a dipole that has been divided in half at its center feed point and fed against a ground plane

Monopole is usually fed from a coaxial cable (see Fig. 8.7 (b))


Monopole is usually fed from a coaxial cable (see Fig. 8.7 (b))

A monopole of length L placed above a perfectly conducting and infinite ground plane will have the same field distribution to that of a dipole of length

2L without the ground plane


Monopole

Ground plane


(b) Monopole of length L over a ground plane

Ground plane

Coaxial cable

Dipole in free space

Image of monopole

8.4 Kinds of antennas This is because an image of the monopole will be formed

inside the ground plane (similar to the method of images in chapter 2)

The monopole looks like a dipole in free space (see Fig. 8.7 (b))


(b))

Since this monopole is of length L only, it will radiate only half of the total radiated power of a dipole of

length 2L

Hence, the radiation resistance of a monopole is half that of a dipole

8.4 Kinds of antennas Similarly, directivity of the monopole is twice that of a dipole

Since the field distributions are the same for a monopole and dipole, the maximum radiation intensity will be also same for both

cases


cases

But for monopole, the total radiated power is half that of a dipole

Hence, the directivity of a monopole above a conducting ground plane is twice that of dipole in free space

8.4 Kinds of antennas8.4.3 Loop antenna

Loop antennas could be of various shapes: circular,

triangular,

square,


square,

elliptical, etc.

They are widely used in applications up to 3GHz

Loop antennas can be classified into two: electrically small (circumference < 0.1 λ) and

electrically large (circumference approximately equals to λ)

8.4 Kinds of antennas Electrically small loop antennas have

very small radiation resistance

They have very low radiation and

are practically useless


are practically useless

Electrically small loop antennas could be analyzed assuming that it is equivalently represented as a Hertz dipole

Let us consider electrically large circular loop of constant current

8.4 Kinds of antennas Fig. 8.9 Loop antenna

θrr


φ

'φ

'P

'rr

ψ

Rr

8.4 Kinds of antennas We can express magnetic vector potential (see textbook) as

( )( )

( )

01 1

2

1 120

( ) ( sin ) ( sin )4

1( ) ( 1) ( ) ( sin ) ( sin )

2 ! !

j r

m m

n n

n n nm nm

I aeA j J a J a

r

zJ z z J z J z J a J a

m n m

β

φ

µθ π β θ β θ

π

β θ β θ

−

∞

+=

= − −

−= ∴ − = − ⇒ − = −

+∑Q


We can express electric field as

( )0

0 1

2 ! !

( sin )( )

2

m

j r

m n m

j I ae J aA

r

β

φ

µ β θθ

=

−

+

∴ =

( )AE j j Aω

ωµε

∇ ∇ •∴ = − −

rrr

8.4 Kinds of antennas Note that magnetic vector potential has only component

which is a function of θ variable onlyφ

0 1

0;

( sin )0 0; 0

j r

r r r

A

I ae J aA A E j A E j A E j A

β

θ θ θ φ φ

ωµ β θω ω ω

−

∴∇ • =

= = ⇒ = − = = − = ∴ = − =

r

Q


0 0; 02

r r rA A E j A E j A E j A

rθ θ θ φ φω ω ω= = ⇒ = − = = − = ∴ = − =Q

( )01 sin ; 0

2

j r

r

E I aeH J a H H

r

βφ

θ φ

ωµβ θ

η η

−−= − = = =

8.4 Kinds of antennas Poynting vector for a wave radiating in radially outward

direction should have direction along positive radial direction

Therefore must be negative Fig. 8.10 shows the far-field radiation pattern of the loop

Hθ


Fig. 8.10 shows the far-field radiation pattern of the loop antenna

It can be observed that the radiation field has higher magnitude with the larger radius of the loop antenna

For larger radiation power we need a loop antenna of larger radius



Fig. 8.10 Plot of for various values of angle θ (far-field radiation patterns) (dotted line a=0.1λ, dashed line a=0.2λ, solid line a=0.3λ)

8.4 Kinds of antennas For small loops

( ) θβθβθβθββ sin2

1...sin

16

1sin

2

1)sin(;

3

1 31 aaaaJa ≅+−=<

( ) ( )0 01 1sin ; sin

j r j rI a I ae eE a H a

β β

φ θ

ωµ ωµβ θ β θ

η

− −

∴ = = −


Note that for the dipole polarization was along direction

But for small loop antennas it is along the direction

( ) ( )sin ; sin2 2 2 2

E a H ar r

φ θβ θ β θη

∴ = = −

θ

φ

8.5 Antenna Arrays

One of the disadvantages of single antenna is that it has fixed radiation pattern

That means once we have designed and constructed an antenna, the beam or radiation pattern is fixed


the beam or radiation pattern is fixed

If we want to tune the radiation pattern, we need to apply the technique of antenna arrays

Antenna array is a configuration of multiple antennas (elements) arranged

to achieve a given radiation pattern

8.5 Antenna Arrays

There are several array design variables which can be changed to achieve the overall array pattern design

Some of the array design variables are:

(a) array shape linear, circular,


circular, planar, etc.

(b) element spacing

(c) element excitation amplitude

(d) element excitation phase

(e) patterns of array elements

φ

8.5 Antenna Arrays

Given an antenna array of identical elements, the radiation pattern of the antenna array may be found

according to the pattern multiplication principle

It basically means that array pattern is equal to φ


It basically means that array pattern is equal to the product of the

pattern of the individual array element into

array factor , a function dependent only on

the geometry of the array

the excitation amplitude and phase of the elements

φ

8.5 Antenna Arrays

8.5.1 Two element array

Let us investigate an array of two infinitesimal dipoles positioned along the z axis as shown in Fig.

8.10 (a)

The field radiated by the two elements, assuming no coupling between the elements


assuming no coupling between the elements

is equal to the sum of the two fields

where the two antennas are excited with current

1 1 2 22 2

1 21 2

1 2

sin sinˆ ˆ4 4

j r j j r j

total

jI dl e e jI dl e eE E E

r r

β δ β δθ β θ βθ θ

πεω πεω

− −

= + = +r r r

1 1 2 2I and Iδ δ< <

8.5 Antenna Arrays

1rr

rr

1θ

θ


Fig. 8.10 (a) Two Hertz dipoles

2rr2θ

8.5 Antenna Arrays

For 1 2 oI I I= =

1 2,2 2

α αδ δ= = −

2 cos cossinˆd dj r j jjI dl e

β α β αβ θ θθβθ

− + − + = +

r


2 cos cos2 2 2 20 sinˆ

4

j r j j

total

jI dl eE e e

r

β θ θθβθ

πεω

− + − +

= +

r

2

0 sinˆ 2cos cos4 2 2

j rjI dl e d

r

βθβ β αθ θ

πεω

− = +

8.5 Antenna Arrays

Hence the total field of the array is equal to

the field of single element positioned at the origin

multiplied by a factor which is called as the array factor

Array factor is given


Normalized array factor is

( )1

2cos cos2

AF dβ θ α

= +

( )2

1cos cos

2AF dβ θ α

= +

8.5 Antenna Arrays

8.5.2 N element uniform linear array (ULA) This idea of two element array can be extended

to N element array of uniform amplitude and spacing

Let us assume that N Hertz dipoles are placed along a straight line along z-axis at positions


straight line along z-axis at positions 0, d, 2d, …, (N-2) d and (N-1) d respectively

8.5 Antenna Arrays Current of equal amplitudes

but with phase difference of 0, α, 2α, … , (N-2) α and (N-1) α


are excited to the corresponding dipoles at 0, d, 2d, …, (N-2) d and (N-1) d respectively

8.5 Antenna Arraysz

2d

(N-1)d

.

.

.

2I α⟨

( 1)I N α⟨ −

0I ⟨

I α⟨

( 1)

2

NI α

−⟨


Fig. 8.10 (b) ULA 1 (c) ULA 2 (assume N is an odd number)

d

2d

00I ⟨

I α⟨

2I α⟨0I ⟨

I α⟨−

( 1)

2

NI α

−⟨−

8.5 Antenna Arrays

Then the array factor for the N element ULA of Fig. 8.10 (b) will become

( ) ( ) ( )cos 2 cos ( 1) cos1 .....

j d j d j N dAF e e e

β θ α β θ α β θ α+ + − +∴ = + + + +

( )( )ψ−1 e

jNN


( )( ) αθβψψ

ψαθβ +=

−

−=⇒=⇒ ∑

=

+− cos;1

1

1

cos1d

e

eAFeAF

j

jN

N

N

n

dnj

1

1

jN

j

e

e

ψ

ψ−

=−

( ) ( )1 2 2( )2

1 1( ) ( )2 2

N Nj jN

j

j j

e ee

e e

ψ ψψ

ψ ψ

−−

−

−=

−

1( )

2

sin( )2

sin( )2

Nj

N

eψ

ψ

ψ

−

=

8.5 Antenna Arrays

If the reference point is at the physical

center of the array as depicted in

Fig. 8.10 (c), the array factor is

( )sin( )

2

Nψ

0I ⟨

I α⟨

( 1)

2

NI α

−⟨


For small values of

( )sin( )

2

sin( )2

NAF

ψ

ψ=

( )sin( )

2

2

N

N

AF

ψ

ψ=

0I ⟨

I α⟨−

( 1)

2

NI α

−⟨−

8.5 Antenna Arrays

The maximum value of AF is for and its value is N

Apply L’ Hospital rule since it is of the form

To normalize the array factor so that the maximum value is equal to unity, we get,

0ψ =

0

0sin


( )sin( )

1 21

sin2

N

N

AFN

ψ

ψ

=

sin( )2

2

N

N

ψ

ψ≅

8.5 Antenna Arrays

This is the normalized array factor for ULA As N increases, the main lobe narrows The number of lobes is equal to N

one main lobe and other N-1 side lobes


other N-1 side lobes

in one period of the AF The side lobes widths are of 2/N and main lobes are two times wider than the side lobes The SLL decreases with increasing N This can be verified from Fig. 8.11 (see textbook)

8.5 Antenna Arrays

Null of the array

To find the null of the array,

( )sin( ) 0 cos2

Ndψ ψ β θ α= = +Q

( )sin( )

1 21

sin2

N

N

AFN

ψ

ψ

=


( )

1

2cos cos

2 2

1 2cos 1,2,3,......

n

N N nn d n d

N

nn

d N

πψ π β θ α π β θ α

πθ α

β−

⇒ = ± ⇒ + = ± ⇒ = − ±

⇒ = − ± =

8.5 Antenna Arrays

Maximum values

It attains the maximum values for 0ψ =

( )1

cos2 2

m

dθ θ

ψβ θ α

=

= + 0=

( )sin( )

1 21

sin2

N

N

AFN

ψ

ψ

=


8.5.3 Broadside array

We know that when

mθ θ=

1cosmd

αθ

β−

−

⇒ =

8.5 Antenna Arrays

the maximum radiation occurs

It is desired that maximum occurs at θ=90˚

c o s 0dψ β θ α= + =

0

0

90cos 0 0d

θψ β θ α α

== + = ⇒ =


8.5.4 Endfire array

We know that when


090cos 0 0d

θψ β θ α α

== + = ⇒ =

cos 0dψ β θ α= + =

8.5 Antenna Arrays

It is desired that maximum occurs at θ=0˚,

8.5.5 Phase scanning array

We know that when

00cos 0d d

θψ β θ α α β

== + = ⇒ = −


We know that when


It is desired that maximum occurs at θ=θ0

cos 0dψ β θ α= + =

00cos 0 cosd d

θ θψ β θ α α β θ

== + = ⇒ = −

8.5 Antenna Arrays

Draw the polar plot of radiation pattern for the following

uniform linear array (ULA)

of N isotropic radiating antennas spaced λ/2

apart for the following cases: θ

I α⟨

( 1)

2

NI α

−⟨


(a) Broadside array (Maximum field is at θ=90˚)

(b) End fire array (Maximum field is at θ=0˚)

(c) Maximum field is at θ=60˚ and

(d) Null at θ=60˚

( )sin( )

1 21

sin2

N

N

AFN

ψ

ψ

=

0I ⟨

I α⟨−

( 1)

2

NI α

−⟨−

8.5 Antenna Arrays 0

0

90cos 0 0d

θψ β θ α α

== + = ⇒ =


(a) Broadside array (Maximum field is at θ=90˚)

8.5 Antenna Arrays 00cos 0d d

θψ β θ α α β

== + = ⇒ = −


(b) End fire array (Maximum field is at θ=0˚)

8.5 Antenna Arrays


(c) Maximum field is at θ=60˚

00cos 0 cosd d

θ θψ β θ α α β θ

== + = ⇒ = −

8.5 Antenna Arrays


(d) Null at θ=60˚L,3,2,1,cos

2=−= nd

N

nnullθβ

πα

8. antennas and radiating systems · 8.1 introduction antennas radiation fundamentals antenna...

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